CAT Quantitative Aptitude Questions | CAT Algebra - Functions questions

The question is about Quadratic Expressions. We need to find the maximum value of a minimum function. Functions is a simple topic which involves a lot of basic ideas. Make sure that you a get of hold of them by solving these questions. A range of CAT questions can be asked based on this simple concept of Functions questions in the CAT Exam. Make sure you practice a lot of questions in CAT Functions Algebra to push your CAT preparation in the right direction. Functions as an idea is a fabulous idea if one can wrap their head around it the right way, and it has been one of the staple ideas tested by the CAT exam historically.

Question 2: Find the maximum value of f(x); if f(x) is defined as the Min {-(x – 1)² + 2, (x – 2)² + 1}

1. 1
2. 2
3. 0
4. 3

Video Explanation

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Functions: The same function can have different functional expressions in different ranges. Finding maximum/minimum values in these cases becomes very interesting.

First let us find the range where Min (-(x – 1)² + 2, (x – 2)² + 1) is – (x – 1)² + 2.
In other words, in which range is – (x – 1)² + 2 < (x – 2)² + 1.
–(x² – 2x +1) + 2 < x² – 4x + 4 + 1
0 < 2x² – 6x + 4
x² – 3x + 2 > 0
(x – 1) (x – 2) > 0
=> x > 2 or x < 1

So, for x ∈ (1, 2) , f(x) = (x – 2)² + 1
And f(x) = –(x – 1)² + 2 elsewhere.

Let us also compute f(1) and f(2)
f(1) = 2, f(2) = 1
For x ∈ (-∞, 1), f(x) = –(x – 1)² + 2
f(1) = 2
For x ∈ (1, 2), f(x) = (x – 2)² + 1
f(2) = 1
For x ∈ (2, ∞), f(x) = –(x – 1)² + 2
For x < 1 and x > 2, f(x) is -(square) + 2 and so is less than 2.

When x lies between 1 and 2, the maximum value it can take is 2. f(1) = 2 is the highest value f(x) can take.
As a simple rule of thumb, the best way to approach this question is to solve the two expressions. This gives us the meeting points of the two curves. One of the two meeting points should be the maximum value.

The question is "Find the maximum value of f(x); if f(x) is defined as the Min {-(x – 1)² + 2, (x – 2)² + 1}?"

Hence the answer is "2"

Choice B is the correct answer.